Branch and Price for Large-Scale Capacitated Hub Location Problems with Single Assignment

This paper presents a branch–and–price algorithm for the Capacitated Hub Location Problem with Single Assignment, in which Lagrangean relaxation is used to obtain tight lower bounds of the Restricted Master Problem (RPM). It is shown how to solve the pricing problem for finding new columns to be added to the RMP, or for proving that the lower bound of the current RMP is valid. Also, a lower bound that is valid at any stage of the column generation algorithm is proposed. The process to obtain this valid lower bound allows to apply a stabilization method that results in a considerable improvement on the overall efficiency of the solution algorithm. Numerical results on a battery of benchmark instances of up to 200 nodes are reported. These seem to be the largest instances that have been solved exactly for this problem.